Volume
Volume is the of enclosed by some closed boundary, for example, the space that a substance ( , , , or ) or shape occupies or contains. Volume is often quantified numerically using the , the . The volume of a is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic s. The volumes of more complicated shapes can be calculated by if a formula exists for the shape's boundary. One-dimensional figures (such as ) and two-dimensional shapes (such as ) are assigned zero volume in the three-dimensional space. The volume of a solid (whether regularly or irregularly shaped) can be determined by . Displacement of liquid can also be used to determine the volume of a gas. The combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the combined volume is not .One liter of sugar (about 970 grams) can dissolve in 0.6 liters of hot water, producing a total volume of less than one liter. In , volume is expressed by means of the , and is an important global . In , volume is a , and is a to . Units . Approximate conversion to milliliters: }} Any unit of gives a corresponding unit of volume, namely the volume of a whose side has the given length. For example, a (cm3) would be the volume of a cube whose sides are one (1 cm) in length. In the (SI), the standard unit of volume is the cubic meter (m3). The also includes the (L) as a unit of volume, where one liter is the volume of a 10-centimeter cube. Thus :1 liter = (10 cm)3 = 1000 cubic centimeters = 0.001 cubic meters, so :1 cubic meter = 100000 liters. Small amounts of liquid are often measured in s, where :1 milliliter = 0.0011 liters = 19 cubic centimeter. Various other traditional units of volume are also in use, including the , the , the , the , the , the , the , the , the , the , the , the , the , the , the , the , and the . Related terms Volume and capacity are sometimes distinguished, with capacity being used for how much a container can hold (with contents measured commonly in s or its derived units), and volume being how much space an object displaces (commonly measured in cubic meters or its derived units). Volume and capacity are also distinguished in , where capacity is defined as volume over a specified time period. However in this context the term volume may be more loosely interpreted to mean quantity. The of an object is defined as mass per unit volume. The inverse of density is which is defined as volume divided by mass. Specific volume is a concept important in where the is often an important parameter of a system being studied. The in is the volume of fluid which passes through a given surface per unit time (for example cubic meters per second s-1). Volume formulas Ratio of volumes of a cone, sphere and cylinder of the same radius and height The above formulas can be used to show that the volumes of a , sphere and of the same radius and height are in the ratio 1 : 2 : 3, as follows. Let the radius be r'' and the height be ''h (which is 2''r'' for the sphere), then the volume of cone is : \tfrac{1}{3} \pi r^2 h = \tfrac{1}{3} \pi r^2 (2r) = (\tfrac{2}{3} \pi r^3) \times 1, the volume of the sphere is : \tfrac{4}{3} \pi r^3 = (\tfrac{2}{3} \pi r^3) \times 2, while the volume of the cylinder is : \pi r^2 h = \pi r^2 (2r) = (\tfrac{2}{3} \pi r^3) \times 3. The discovery of the 2 : 3 ratio of the volumes of the sphere and cylinder is credited to . Volume formula derivations Sphere The volume of a is the of infinitesimal circular slabs of thickness dx. The calculation for the volume of a sphere with center 0 and radius r'' is as follows. The surface area of the circular slab is \pi r^2 . The radius of the circular slabs, defined such that the x-axis cuts perpendicularly through them, is; y = \sqrt{r^2-x^2} or z = \sqrt{r^2-x^2} where y or z can be taken to represent the radius of a slab at a particular x value. Using y as the slab radius, the volume of the sphere can be calculated as \int_{-r}^r \pi y^2 \,dx = \int_{-r}^r \pi(r^2 - x^2) \,dx. Now \int_{-r}^r \pi r^2\,dx - \int_{-r}^r \pi x^2\,dx = \pi (r^3 + r^3) - \frac{\pi}{3}(r^3 + r^3) = 2\pi r^3 - \frac{2\pi r^3}{3}. Combining yields gives V = \frac{4}{3}\pi r^3. This formula can be derived more quickly using the formula for the sphere's , which is 4\pi r^2 . The volume of the sphere consists of layers of infinitesimal spherical slabs, and the sphere volume is equal to \int_0^r 4\pi u^2 \,du = \frac{4}{3}\pi r^3. Cone The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies cones as well. But for an explanation using calculus: The volume of a is the of infinitesimal circular slabs of thickness ''dx. The calculation for the volume of a cone of height h'', whose base is centered at (0,0,0) with radius ''r, is as follows. The radius of each circular slab is r'' if ''x = 0 and 0 if x'' = ''h, and varying linearly in between—that is, r\frac{(h-x)}{h}. The surface area of the circular slab is then \pi \left(r\frac{(h-x)}{h}\right)^2 = \pi r^2\frac{(h-x)^2}{h^2}. The volume of the cone can then be calculated as \int_{0}^h \pi r^2\frac{(h-x)^2}{h^2} dx, and after extraction of the constants: \frac{\pi r^2}{h^2} \int_{0}^h (h-x)^2 dx Integrating gives us \frac{\pi r^2}{h^2}\left(\frac{h^3}{3}\right) = \frac{1}{3}\pi r^2 h. See also * * * * * * * * * * * * * References External links *Volume calculator - Javascript automatic calculator. Category:Measurable quantities